Geodesic
THE C 1-Invariance of the Godbillon-Vey Map in Analytical K-Theory
Canadian journal of mathematics, Tome 39 (1987) no. 5, pp. 1210-1222

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An action α of a discrete group Γ on the circle S 1 as orientation preserving C ∞-diffeomorphisms gives rise to a foliation on the homotopy quotient S 1Γ, and its Godbillon-Vey invariant is, by definition, a cohomology class of S 1Γ([1]). This cohomology class naturally defines an additive map from the geometric K-group K 0(S 1, Γ) into C, through the Chern character from K 0(S 1, Γ) to H *(S 1, Γ Q).Using cyclic cohomology, Connes constructed in [2] an additive map, GV(α), which we shall call the Godbillon-Vey map, from the K 0-group of the reduced crossed product C*-algebra C(S 1) ⋊ αΓ into C. He showed that GV(α) agrees with the geometric Godbillon-Vey invariant through the index map μ from K 0(S 1, Γ) to K 0(C(S 1) ⋊ αΓ). 
Natsume, Toshikazu. THE C 1-Invariance of the Godbillon-Vey Map in Analytical K-Theory. Canadian journal of mathematics, Tome 39 (1987) no. 5, pp. 1210-1222. doi: 10.4153/CJM-1987-061-8
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